nLab supergravity Lie 6-algebra



\infty-Lie theory

∞-Lie theory (higher geometry)


Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids



Related topics


\infty-Lie groupoids

\infty-Lie groups

\infty-Lie algebroids

\infty-Lie algebras

String theory




The supergravity Lie 6-algebra (D’Auria & Fré 1982, p. 18) is (with hindsight, following FSS15) a super L-∞ algebra such that L L_\infty -algebra valued differential forms (local ∞-connections) with values in it encode field histories of

  1. the vielbein field with a spin connection,

    hence the first order formulation of gravity for a graviton field

    in 10+1 dimensions;

  2. the gravitino field;

  3. the supergravity C-field

    and its magnetic dual.

This is such that the field strengths and Bianchi identities of these fields are governed by certain fermionic super L-∞ algebraic cocycles as suitable for 11-dimensional supergravity.


Our normalization conventions entirely follow Castellani, D’Auria & Fré 1991, §III.8.3, but we denote the super-vielbein 1-forms by {e a} a=0 10\big\{e^a\big\}_{a=0}^{10} instead of by {V a} a=0 10\big\{V^a\big\}_{a=0}^{10}.


The supergravity Lie 3-algebra 𝔰𝔲𝔤𝔯𝔞 3(10,1)\mathfrak{sugra}_3(10,1) carries an L-∞ algebra cocycle μ 7CE(𝔰𝔲𝔤𝔯𝔞 3(10,1))\mu_7 \in CE\big(\mathfrak{sugra}_3(10,1)\big) of degree 7, given in the standard generators {e a}\{e^a\} (vielbein), {ω ab}\{\omega^{a b}\} (spin connection) {ψ α}\{\psi^\alpha\} (gravitino) and {c 3}\{c_3\} (supergravity C-field) by

μ 7i2ψ¯Γ a 1a 5ψe a 1e a 5+15μ 4c 3, \mu_7 \;\coloneqq\; \frac{i}{2}\bar \psi \wedge \Gamma^{a_1 \cdots a_5} \psi \wedge e_{a_1} \wedge \cdots \wedge e_{a_5} + 15 \mu_4 \wedge c_3 \,,


μ 4=12ψ¯Γ a 1a 2ψe a 1e a 2 \mu_4 \;=\; \frac{1}{2} \overline{\psi} \Gamma^{a_1 a_2} \psi \wedge e_{a_1} \wedge e_{a_2}

is the 4-cocycle which defines 𝔰𝔲𝔤𝔯𝔞 3(10,1)\mathfrak{sugra}_3(10,1) as an extension of 10,1|32\mathbb{R}^{10,1\vert \mathbf{32}}, and where c 3c_3 is the generator that cancels the class of this cocycle, d CEc 3=μ 4d_{CE} c_3 \,=\, \mu_4.

In other words, we have

dμ 4=0 dμ 7=15μ 4μ 4. \begin{array}{l} \mathrm{d} \, \mu_4 \;=\; 0 \\ \mathrm{d} \, \mu_7 \;=\; 15\, \mu_4 \wedge \mu_4 \mathrlap{\,.} \end{array}

which has the same structure as the equations of motion of the field strength G 4G_4 of the supergravity C-field and its Hodge dual G 7=*G 4G_7 = \ast G_4 in 11-dimensional supergravity (to which it is related below)

This appears (in the dual language of Chevalley-Eilenberg algebras) in DAuria & FrFré 1982, page 18 and Castellani, D’Auria & Fré 1991, §III.8.3.


One computes

d CEμ 7= 54ψ¯Γ a 1a 4bψe a 1e a 4ψ¯Γ bψ i15Γ abe aψ¯Γ bψc 3 +154ψ¯Γ abψe ae bψ¯Γ cdψe ce d. \begin{aligned} d_{{}_{CE}} \mu_7 \;=\; & - \frac{5}{4} \bar \psi \wedge \Gamma^{a_1 \cdots a_4 b} \psi \wedge e_{a_1} \wedge \cdots \wedge e_{a_4} \wedge \bar \psi \wedge \Gamma_b \psi \\ & - i 15 \wedge \Gamma^{a b} e_a \wedge \bar \psi \wedge \Gamma_b \psi \wedge c_3 \\ & + \frac{15}{4} \bar \psi \wedge \Gamma_{a b} \psi \wedge e^a \wedge e^b \wedge \bar \psi \wedge \Gamma_{c d} \psi \wedge e^c \wedge e^d \end{aligned} \,.

This expression vanishes due to the Fierz identities

ψ¯Γ a 1a 4bψψ¯Γ bψ=3ψ¯Γ [a 1a 2ψψ¯Γ a 3a 4]ψ \bar \psi \wedge \Gamma^{a_1 \cdots a_4 b} \psi \wedge \bar \psi \wedge \Gamma_b \psi \;=\; 3 \bar \psi \wedge \Gamma^{[a_1 a_2} \psi \wedge \bar \psi \wedge \Gamma^{a_3 a_4 ]} \psi


ψ¯Γ abψψ¯Γ bψ=0. \bar \psi \wedge \Gamma^{a b} \psi \wedge \bar \psi \wedge \Gamma_b \psi \;=\; 0 \,.

The supergravity Lie 6-algebra 𝔰𝔲𝔤𝔯𝔞 7(10,1)\mathfrak{sugra}_{7}(10,1) is the super Lie 7-algebra that is the b 6b^6 \mathbb{R}-extension of 𝔰𝔲𝔤𝔯𝔞 3(10,1)\mathfrak{sugra}_3(10,1) classified by the cocycle μ 7\mu_7 from def. .

b 5𝔰𝔲𝔤𝔯𝔞 6𝔰𝔲𝔤𝔯𝔞 3. b^5 \mathbb{R} \to \mathfrak{sugra}_6 \to \mathfrak{sugra}_3 \,.

This means that the Chevalley-Eilenberg algebra CE(𝔰𝔲𝔤𝔯𝔞 6)CE(\mathfrak{sugra}_6) is generated from

with differential defined by

ω ab ω acω c b e a ω abe b12iψ¯Γ aψ ψ 14ω abΓ ab c 3 12ψ¯Γ abψe ae b c 6 12ψ¯Γ a 1a 5ψe a 1e a 5132ψ¯Γ a 1a 2ψe a 1e a 2c 3. \begin{array}{ccl} \omega^{a b} &\mapsto& \omega^{a c} \wedge \omega_c{}^b \\ e^a &\mapsto& -\omega^{a b} e_b - \frac{1}{2}i \bar \psi \wedge \Gamma^a \psi \\ \psi &\mapsto& - \frac{1}{4}\omega^{a b} \Gamma^{a b} \\ c_3 &\mapsto& \frac{1}{2} \bar \psi \wedge \Gamma^{a b} \psi \wedge e_a \wedge e_b \\ c_6 &\mapsto& - \frac{1}{2} \bar \psi \wedge \Gamma^{a_1 \cdots a_5} \psi \wedge e_{a_1} \wedge \cdots \wedge e_{a_5} - \frac{13}{2} \bar \psi \Gamma^{a_1 a_2} \psi \wedge e_{a_1} \wedge e_{a_2} \wedge c_3 \,. \end{array}

This appears in Castellani, D’Auria & Fré, (III.8.18).


According to Castellani, D’Auria & Fré 1991, comment below (III.8.18): “no further extension is possible”.

Relation to D=11D = 11 supergravity

The supergravity Lie 6-algebra is something like the gauge L L_\infty-algebra of 11-dimensional supergravity, in the sense discussed at D'Auria-Fré formulation of supergravity .


Write W(𝔰𝔲𝔤𝔯𝔞 6(10,1))W\big(\mathfrak{sugra}_6(10,1)\big) for the Weil algebra of the supergravity Lie 6-algebra.

Write g 4g_4 and g 7g_7 for the shifted generators of the Weil algebra corresponding to c 3c_3 and c 6c_6, respectively.

Define an adjusted Weil algebra W˜(𝔰𝔲𝔤𝔯𝔞 6(10,1))\tilde W(\mathfrak{sugra}_6(10,1)) by declaring it to have the same generators and differential as before, except that the differential for c 6c_6 is modified to

d W˜c 6d Wc 6+15g 4c 3 d_{\tilde W} \, c_6 \coloneqq d_{W} \, c_6 + 15 \, g_4 \wedge c_3

and hence the differential of g 7g_7 is accordingly modified in the unique way that ensures d W˜ 2=0d_{\tilde W}^2 = 0 (yielding the modified Bianchi identity for g 7g_7).

This ansatz appears in Castellani, D’Auria & Fré 1991 (III.8.24).

Note that this amounts simply to a redefinition of curvatures

g˜ 7g 7+15g 4c 3. \tilde g_7 \;\coloneqq\; g_7 + 15 g_4 \wedge c_3 \,.


A field configuration of 11-dimensional supergravity is given by L-∞ algebra valued differential forms with values in 𝔰𝔲𝔤𝔯𝔞 6\mathfrak{sugra}_6. Among all of these the solutions to the equations of motion (the points in the covariant phase space) can be characterized as follows:

A field configuration

Ω (X)W˜(𝔰𝔲𝔤𝔯𝔞 6):Φ \Omega^\bullet(X) \leftarrow \tilde W(\mathfrak{sugra}_6) \;\colon\; \Phi

solves the equations of motion precisely if

  1. all curvatures sit in the ideal of differential forms spanned by the 1-form fields E aE^a (vielbein) and Ψ\Psi (gravitino);

    more precisely if we have

    • τ=0\tau = 0


    • G 4=(G 4) a 1,a 4E a 1E a 4G_4 = (G_4)_{a_1, \cdots a_4} E^{a_1} \wedge \cdots E^{a_4}

      (field strength of supergravity C-field)

    • G 7=(G 7) a 1,a 7E a 1E a 7G_7 = (G_7)_{a_1, \cdots a_7} E^{a_1} \wedge \cdots E^{a_7}

      (dual field strength)

    • ρ=ρ abE aE b+H aΨE a\rho = \rho_{a b} E^a \wedge E^b + H_a \Psi \wedge E^a

      (Dirac operator applied to gravitino)

    • R ab=R cd abE cE d+Θ¯ ab cΨE c+Ψ¯K abΨR^{a b} = R^{a b}_{c d} E^c \wedge E^d + \bar \Theta^{a b}{}_c \Psi \wedge E^c + \bar \Psi \wedge K^{a b} \Psi

      (Riemann tensor: field strength of gravity)

  2. such that the coefficients of terms containing Ψ\Psis are polynomials in the coefficients of the terms containing no Ψ\Psis. (“rheonomy”).

This is, in paraphrase, the content of Castellani, D’Auria & Fré, section III.8.5.


In particular, the Bianchi identity for the super-form enhancement of the supergravity C-field flux density is (by III.8.23j, 34, 35, 36 and using hupf):

(1)d(F a 1a 4e a 1e a 4+iψ¯Γ a 1a 5ψe a 1e a 5G 7 super) =15(F a 1a 4e a 1e a 4+12ψ¯Γ a 1a 2ψe a 1e a 2G 4 super)(F a 1a 4e a 1e a 4+12ψ¯Γ a 1a 2ψe a 1e a 2G 4 super) \begin{array}{l} \mathrm{d} \big( \underset{ G_7^{super} }{ \underbrace{ F_{a_1 \cdots a_4} e^{a_1} \wedge \cdots \wedge e^{a_4} \,+\, \mathrm{i} \overline{\psi} \Gamma_{a_1 \cdots a_5} \psi \, e^{a_1} \wedge \cdots \wedge e^{a_5} } } \big) \\ \;=\; -15 \big( \underset{G_4^{super}}{ \underbrace{ F_{a_1 \cdots a_4} e^{a_1} \wedge \cdots \wedge e^{a_4} + \tfrac{1}{2} \overline{\psi}\Gamma_{a_1 a_2}\psi \, e^{a_1} \wedge e^{a_2} } } \big) \wedge \big( \underset{G_4^{super}}{ \underbrace{ F_{a_1 \cdots a_4} e^{a_1} \wedge \cdots \wedge e^{a_4} + \tfrac{1}{2} \overline{\psi}\Gamma_{a_1 a_2}\psi \, e^{a_1} \wedge e^{a_2} } } \big) \mathrlap{\,} \end{array}

and its rheonomic solution implies for the “actual” flux densities F 4F a 1a 4e a 1e a 4| ψ=0F_4 \coloneqq F_{a_1 \cdots a_4} e^{a_1} \wedge \cdots \wedge e^{a_4}\vert_{\psi=0} and G 4G a 1a 7e a 1e a 7| ψ=0G_4 \coloneqq G_{a_1 \cdots a_7} e^{a_1} \wedge \cdots e^{a_7} \vert_{\psi = 0} that

  1. the CJS higher Maxwell equation holds

    dG 7=15F 4F 4 \mathrm{d}\, G_7 \;=\; - 15 \, F_4 \wedge F_4


  2. the Hodge duality holds

    G 7=F 4 G_7 \;=\; \star F_4


supergravity Lie 6-algebra \to supergravity Lie 3-algebra \to super Poincaré Lie algebra


The supergravity Lie 6-algebra originates in:

Brief review is in:

A monograph account is in:

It was (apparently) rediscovered in:

where a detailed discussion is given.

These authors all consider the Chevalley-Eilenberg algebra (“FDA”) of the actual super L L_\infty -algebra. The latter and its relation to smooth super ∞-groupoids is considered in:

Last revised on February 16, 2024 at 11:43:29. See the history of this page for a list of all contributions to it.